Finite Group DatabasebetaFinite Group Databasebeta| order | 7488 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C24×C3 |
| direct product | C22⋊A4×AGL(1,𝔽13) |
| Fitting, socle, derived | C24×C13 ⋊ C32×C4 |
| Hall | C24×#261 ⋊ C32 |
| Sylow | C13 ⋊ C3×C4×C22⋊A4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C24×C13 | C32×C4 |
| prime radical | C22⋊A4×D13⋊C3 | C2 |
| cosocle | C24×D13 | C2×C32 |
| LCS | G ▷ C24×C13 | |
| UCS | C1 | |
1132 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 1132 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 156 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,9,3)(4,5,12,10)(6,11,13,8)(15,16,17)(19,20,21);(2,4,9)(3,7,13)(5,10,11)(6,12,8)(18,21)(19,20);(14,17)(15,16)104 conjugacy classes — view the full character table.
| maximal subgroup of | A42×AGL(1,𝔽13), #1020×AGL(1,𝔽13) |
| maximal normal subgroup of | A42×AGL(1,𝔽13), #1020×AGL(1,𝔽13) |
| maximal quotient of | #997×AGL(1,𝔽13) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,9,3)(4,5,12,10)(6,11,13,8)(15,16,17)(19,20,21), (2,4,9)(3,7,13)(5,10,11)(6,12,8)(18,21)(19,20), (14,17)(15,16));